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# Mathematics and Statistics

Program | Faculty | Master's | Doctoral | Courses

All courses carry 3 credits unless otherwise specified.

## Mathematics

503 Topics in Computer-Connected Mathematics for Secondary Teachers
Computer-connected treatment of many topics appropriate for secondary school mathematics courses using computers, focusing on algorithms, round off error, and graphics. Topics chosen from number theory, linear algebra, geometry, analysis, probability, and statistics. Prerequisites: MATH 233 and 235 or equivalents; working knowledge of Basic, PASCAL, or Fortran.

511 Abstract Algebra I
Introduction to various topics in abstract algebra, such as groups, rings, and fields. A deeper and more advanced treatment than MATH 411. Prerequisites: MATH 235; MATH 300 or consent of instructor.

512 Abstract Algebra II
A continuation of MATH 511.

513 Combinatorics and Graph Theory
Cross-listed with CMPSCI 575. A basic introduction to combinatorics and graph theory for advanced students in computer science, mathematics, and related fields. Topics include elements of graph theory, Euler and Hamiltonian circuits, graph coloring, matching, basic counting methods, generating functions, recurrences, inclusion-exclusion, Polya’s theory of counting. Prerequisites: mathematical maturity, calculus, linear algebra, discrete mathematics course such as CMPSCI 250 or MATH 455. MATH 411 recommended but not required.

523 Introduction to Modern Analysis
Construction of the real number system; sequences, series, functions of one real variable, limits, continuity, differentiability, Riemann integral; sequences and series of functions. Prerequisites: MATH 233, 235 and 300 or equivalent.

532 Topics in Ordinary Differential Equations
Topics chosen from: Sturm-Liouville theory, series solutions, stability theory and singular points, numerical methods, transform methods. Prerequisite: MATH 235 and 431.

534 Introduction to Partial Differential Equations
Classification of second-order partial differential equations, wave equation, Laplace’s equation, heat equation, separation of variables. Prerequisites: MATH 233, 235, and 431.

545 Linear Algebra for Applied Mathematics
Basic concepts (over real or complex numbers): vector spaces, basis, dimension, linear transformations and matrices, change of basis, similarity. Study of a single linear operator: minimal and characteristic polynomial, eigenvalues, invariant subspaces, triangular form, Cayley-Hamilton theorem. Inner product spaces and special types of linear operators (over real or complex fields): orthogonal, unitary, self-adjoint, hermitian. Diagonalization of symmetric matrices, applications. Prerequisite: MATH 235 or equivalent.

551 Numerical Analysis I
Introduction to computational techniques used in science and industry. Topics selected from root-finding, interpolation, data fitting, linear systems, numerical integration, numerical solution of differential equations, and error analysis. Prerequisites: MATH 233, and either MATH 235 or consent of instructor; knowledge of a high level programming language.

552 Numerical Analysis II
Introduction to the application of computational methods to models arising in science and engineering, focusing mainly on the solution of partial differential equations. Topics include finite differences, finite elements, boundary value problems, fast Fourier transforms. Prerequisite: MATH 551 or consent of instructor.

563 Differential Geometry
Differential geometry of curves and surfaces in Euclidean 3-space using vector methods. Prerequisites: MATH 233 and 235.

611 Algebra I
Introduction to groups, rings, and fields. Direct sums and products of groups, cosets, Lagrange’s theorem, normal subgroups, quotient groups. Polynomial rings, UFDs and PIDs, division rings. Fields of fractions, GCD and LCM, irreducibility criteria for polynomials. Prime field, characteristic, field extension, finite fields. Some topics from MATH 612 included.

612 Algebra II

A continuation of MATH 611. Topics in group theory (e.g., Sylow theorems, solvable and simple groups, Jordan-Holder and Schreier theorems, finitely generated Abelian groups). Topics in ring theory (matrix rings, prime and maximal ideals, Noetherian rings, Hilbert basis theorems). Modules, including cyclic, torsion, and free modules, direct sums, tensor products. Algebraic closure of fields, normal, algebraic, and transcendental field extensions, basic Galois theory. Prerequisite: MATH 611 or equivalent.

621 Complex Analysis
Complex number field, elementary functions, holomorphic functions, integration, power and Laurent series, harmonic functions, conformal mappings, applications.

623 Real Analysis I
General theory of measure and integration and its specialization to Euclidean spaces and Lebesgue measure; modes of convergence, Lp spaces, product spaces, differentiation of measures and functions, signed measures, Radon-Nikodym theorem.

624 Real Analysis II
Continuation of MATH 623. Introduction to functional analysis; elementary theory of Hilbert and Banach spaces; functional analytic properties of Lp-spaces, applications to Fourier series and integrals; interplay between topology, and measure, Stone-Weierstrass theorem, Riesz representation theorem. Further topics depending on instructor.

645 Differential Equations and Dynamical Systems I
Classical theory of ordinary differential equations and some of its modern developments in dynamical systems theory. Linear systems and exponential matrix solutions. Well-posedness for nonlinear systems. Qualitative theory: limit sets, invariant set and manifolds. Stability theory: linearization about an equilibrium, Lyapunov functions. Autonomous two-dimensional systems and other special systems. Prerequisites: advanced calculus, linear algebra and basic ODE.

646 Differential Equations and Dynamical Systems II
A continuation of MATH 645. Topics in the theory of dynamical systems, possibly including: Floquet theory of periodic solutions; bifurcation theory; complex behavior of higher dimensional systems, attractors and chaos. Applications from physical and biological sciences illustrate and motivate the theoretical material. Prerequisite: MATH 645.

651 Numerical Analysis I
The analysis and application of the fundamental numerical methods used to solve a common body of problems in science. Linear system solving: direct and iterative methods. Interpolation of data by function. Solution of nonlinear equations and systems of equations. Numerical integration techniques. Solution methods for ordinary differential equations. Emphasis on computer representation of numbers and its consequent effect on error propagation. Prerequisites: advanced calculus, knowledge of a scientific programming language.

652 Numerical Analysis II
Presentation of the classical finite difference methods for the solution of the prototype linear partial differential equations of elliptic, hyperbolic, and parabolic type in one and two dimensions. Finite element methods developed for two dimensional elliptic equations. Major topics include consistency, convergence and stability, error bounds, and efficiency of algorithm. Prerequisites: MATH 651, familiarity with partial differential equations.

671 Topology
Topological spaces. Metric spaces. Compactness, local compactness. Product and quotient topology. Separation axioms. Connectedness. Function spaces. Fundamental group and covering spaces.

696 Independent Study
Credit, 1-6.

699 Master’s Thesis

701, 702 Topics in Algebra
Consent of instructor required. Credit, 1-3 each semester.

703 Theory of Manifolds I
Inverse and implicit functions theorems, rank of a map. Regular and critical values. Sard’s theorem. Differentiable manifolds, submanifolds, embeddings, immersions and diffeomorphisms. Tangent space and bundle, differential of a map. Partitions of unity, orientation, transversality embed-dings in Rn. Vector fields, local flows. Lie bracket, Frobenius theorem. Lie groups, matrix Lie groups, left invariant vector fields. Prerequisite: MATH 671 or consent of instructor.

704 Theory of Manifolds II
Tensor calculus, differential forms, integration, closed and exact forms. DeRham cohomology. Vector bundles. Introduction to Riemannian Geometry: Gaussian curvature, curvature tensor. Connections, curvature, Chern classes. Prerequisite: MATH 703.

705, 706 Topics in Analysis
Consent of instructor required. Credit, 1-3 each semester.

Consent of instructor required. Credit, 3 each semester.

713 Introduction to Algebraic Number Theory
Valuations, rings of integral elements, ideal theory in algebraic number fields of algebraic functions of one variable, Dirichlet-Hasse unit theorem and Riemann-Roch theorem for curves. Prerequisites: MATH 611 and 612 or equivalent.

731 Introduction to Partial Differential Equations I
Introduction to the modern methods in partial differential equations. Calculus of distributions: weak derivatives, mollifiers, convolutions and Fourier transform. Prototype linear equations of hyperbolic, parabolic and elliptic type, and their fundamental solutions. Initial value problems: Cauchy problem for wave and diffusion equations; well-posedness in the Hilbert-Sobolev setting. Boundary value problems: Dirichlet and Neumann problems for Laplace and Poisson equations; variational formulation and weak solutions; basic regularity theory; Green functions and operators; eigenvalue problems and spectral theorem. Prerequisites: advanced calculus and MATH 623-624.

732 Introduction to Partial Differential Equations II
A continuation of MATH 731. General class of equations and systems, modeled on the prototypes studied in 731. Linear hyperbolic systems. Parabolic evolution equations, and semigroups of operators. Linear elliptic equations of second order. Topics in nonlinear equations. Possible topics include: hyperbolic conservation laws; nonlinear parabolic systems—reaction-diffusion equations, Navier-Stokes equations; mean curvature equations; free-boundary problems. Prerequisite: MATH 731.

781 Algebraic Topology I
Homotopy theory, simplicial and Cech homology theories. Prerequisites: MATH 611, 671.

782 Algebraic Topology II
General homology theory, universal co-efficient theorems, singular homology theories, ring structure in cohomology theories, spectral sequences. Steenrod operations. Prerequisite: MATH 781.

796, 896 Independent Study
Maximum credit, 6.

Credit, 3 each semester.

892, 893, 894, 895 Research Seminar
Presentation by advanced graduate students of research articles, perhaps own research. Credit, 1 each semester.

899 Doctoral Dissertation
Credit, 18.

## Statistics

501 Methods of Applied Statistics
For graduate and upper-level undergraduate students, with focus on practical aspects of statistical methods. Topics include: data description and display, probability, random variables, random sampling, estimation and hypothesis testing, one and two sample problems, analysis of variance, simple and multiple linear regression, contingency tables. Includes data analysis using a computer package. Prerequisites: high school algebra; junior standing or higher.

505 Regression Analysis
Inferences in simple and multiple regression models, model fitting, checking and selection, diagnostics, presentation of the multiple linear regression model in matrix form. Has a strong applied component involving the use of a statistical package for data analysis. Prerequisite: previous coursework in statistics.

506 Design of Experiments
Planning, statistical analysis and interpretation of experiments. Designs considered include factorial designs, randomized blocks, latin squares, incomplete balanced blocks, nested and crossover designs, mixed models. Has a strong applied component involving the use of a statistical package for data analysis. Prerequisite: previous coursework in statistics.

511 Multivariate Statistical Methods
Introduction to the analysis of multivariate data. Topics include description of multivariate data; random vectors; multivariate analysis of variance, repeated measures/profile analysis; and topics from multivariate regression, discriminant analysis, clustering, (principal components, factor analysis, and canonical correlation). Has a strong applied component involving the use of a statistical package for data analysis. Prerequisite: previous background in statistics, or consent of instructor.

515 Introduction to Statistics I
First semester of a two-semester sequence. Emphasis on those parts of probability theory necessary for statistical inference. Probability models, sample spaces, conditional probability, independence, random variables, expectation, variance, discrete and continuous probability distributions, joint distributions, sampling distributions, the central limit theorem. Prerequisites: MATH 131, 132.

516 Introduction to Statistics II
Basic ideas of point and interval estimation and hypothesis testing; one and two sample problems, simple linear regression, topics from among one-way analysis of variance, discrete data analysis and nonparametric methods. Prerequisite: STATISTC 515 or equivalent.

605 Probability Theory
A modern treatment of probability theory based on abstract measure and integration. Random variables, expectations, independence, laws of large numbers, central limit theorem, and general conditioning using the Radon-Nikodym theorem. Introduction to stochastic processes: martingales, Brownian motion. Prerequisite: MATH 623.

607 Mathematical Statistics I
Probability theory, including random variables, independence, laws of large numbers, central limit theorem; statistical models; introduction to point estimation, confidence intervals, and hypothesis testing. Prerequisite: advanced calculus and linear algebra, or consent of instructor.

608 Mathematical Statistics II
Point and interval estimation, hypothesis testing, large sample results in estimation and testing; decision theory; Bayesian methods; analysis of discrete data. Also, topics from nonparametric methods, sequential methods, regression, analysis of variance. Prerequisite: STATISTC 607 or equivalent.

640 Sampling Theory
Introduction to the theory and practice of sampling from finite populations. Designs covered include simple random sampling, systematic sampling, cluster sampling, stratified sampling, sampling with unequal probabilities, multistage and double sampling. Also, ratio, regression and jackknife estimators and the determination of samples sizes. Prerequisite: calculus-based background in probability and statistics.

705 Linear Models I
First semester of two-semester sequence in the theory of linear models. Basic results on the multivariate normal distribution; linear and quadratic forms; noncentral Chi-square and F distributions; inference in linear models, including point and interval estimation, hypothesis testing, etc. Prerequisites: STATISTC 607-608 or equivalent; linear algebra.

706 Linear Models II
Second semester of sequence in theory of linear models with focus on “analysis of variance/design of experiments” models. Includes factorial experiments (balanced and unbalanced designs, notions of interaction, etc.), randomized block designs, incomplete designs (incomplete block designs and latin squares), random effects, nested models, and mixed models.

708 Applied Stochastic Models and Methods
Stochastic processes and their applications to engineering, the physical and social sciences, and finance. General topics: discrete-time processes (e.g., Markov chains, stationary sequences); continuous-time processes (e.g., Markov, Gaussian); spatial processes (e.g., Markov random fields). Specific applications, depending on the composition of the class may include: statistical physics, queuing theory, neural networks, time series prediction, stochastic algorithms in optimization, image processing, stochastic differential equations in finance, control theory. Prerequisites: STATISTC 607-608.

712 Multivariate Analysis
The fundamentals of distribution theory, estimation and hypothesis testing in the multivariate framework. Includes multivariate regression, multivariate anova, discriminant analysis/classification, principal components, factor analysis, cluster analysis and canonical correlation. Prerequisites: STATISTC 607-608.

725-726 Estimation Theory and Hypothesis Testing
The advanced theory of statistics, including methods of estimation (unbiasedness, equivariance, maximum likelihood, Bayesian, minimax), optimality properties of estimators, hypothesis testing, uniformly most powerful tests, unbiased tests, invariant tests, relationship between confidence regions and tests, large sample properties of tests and estimators. Prerequisites: STATISTC 605 and 608.

741, 742 Recent Developments in Statistics
Content varies with instructor. Possible topics include nonparametric statistics, reliability/survival analysis, time series, generalized linear models.

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